# Convex solutions to the mean curvature flow

### Abstract

In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\Bbb{R}^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a sphere or cylinder $S^{k-1}\times \Bbb{R}^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension $2$. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is $k$-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space $\Bbb{R}^{n+1}$ is a shrinking sphere or cylinder.

• [1] B. Andrews, "Contraction of convex hypersurfaces in Euclidean space," Calc. Var. Partial Differential Equations, vol. 2, iss. 2, pp. 151-171, 1994.
@article {1, MRKEY = {1385524},
AUTHOR = {Andrews, Ben},
TITLE = {Contraction of convex hypersurfaces in {E}uclidean space},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {2},
YEAR = {1994},
NUMBER = {2},
PAGES = {151--171},
ISSN = {0944-2669},
MRCLASS = {53A07 (35K55 58G11)},
MRNUMBER = {1385524},
MRREVIEWER = {John Urbas},
DOI = {10.1007/BF01191340},
ZBLNUMBER = {0805.35048},
}
• [2] S. Bernstein, "Sur un théorème de géométrie et son application aux equations aux dérivées partielles du type elliptique," Comm. Soc. Math. Kharkvo, vol. 15, pp. 38-45, 1915.
@article{2,
author= {Bernstein, Serge},
TITLE = {Sur un théorème de géométrie et son application aux equations aux dérivées partielles du type elliptique},
JOURNAL={Comm. Soc. Math. Kharkvo},
VOLUME={15},
YEAR={1915},
PAGES={38--45},
}
• [2b] S. Bernstein, "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus," Math. Z., vol. 26, iss. 1, pp. 551-558, 1927.
@article {2b, MRKEY = {1544873},
AUTHOR = {Bernstein, Serge},
TITLE = {Über ein geometrisches {T}heorem und seine {A}nwendung auf die partiellen {D}ifferentialgleichungen vom elliptischen {T}ypus},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {26},
YEAR = {1927},
NUMBER = {1},
PAGES = {551--558},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {Contributed Item},
MRNUMBER = {1544873},
DOI = {10.1007/BF01475472},
}
• [3] Y. G. Chen, Y. Giga, and S. Goto, "Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations," J. Differential Geom., vol. 33, iss. 3, pp. 749-786, 1991.
@article {3, MRKEY = {1100211},
AUTHOR = {Chen, Yun Gang and Giga, Yoshikazu and Goto, Shun'ichi},
TITLE = {Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {33},
YEAR = {1991},
NUMBER = {3},
PAGES = {749--786},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {35K65 (35D05 53C21 58E12)},
MRNUMBER = {1100211},
MRREVIEWER = {Ioan I. Vrabie},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214446564},
ZBLNUMBER = {0696.35087},
}
• [4] K. Chou and X. Wang, "Entire solutions of the Monge-Ampère equation," Comm. Pure Appl. Math., vol. 49, iss. 5, pp. 529-539, 1996.
@article {4, MRKEY = {1377561},
AUTHOR = {Chou, Kai-Seng and Wang, Xu-Jia},
TITLE = {Entire solutions of the {M}onge-{A}mpère equation},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {49},
YEAR = {1996},
NUMBER = {5},
PAGES = {529--539},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {35J60},
MRNUMBER = {1377561},
MRREVIEWER = {John Urbas},
DOI = {10.1002/(SICI)1097-0312(199605)49:5<529::AID-CPA2>3.3.CO;2-G},
ZBLNUMBER = {0851.35035},
}
• [5] K. Ecker and G. Huisken, "Mean curvature evolution of entire graphs," Ann. of Math., vol. 130, iss. 3, pp. 453-471, 1989.
@article {5, MRKEY = {1025164},
AUTHOR = {Ecker, Klaus and Huisken, Gerhard},
TITLE = {Mean curvature evolution of entire graphs},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {130},
YEAR = {1989},
NUMBER = {3},
PAGES = {453--471},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {53A10 (53C45)},
MRNUMBER = {1025164},
MRREVIEWER = {S. Walter Wei},
DOI = {10.2307/1971452},
ZBLNUMBER = {0696.53036},
}
• [6] L. C. Evans, "Classical solutions of fully nonlinear, convex, second-order elliptic equations," Comm. Pure Appl. Math., vol. 35, iss. 3, pp. 333-363, 1982.
@article {6, MRKEY = {0649348},
AUTHOR = {Evans, Lawrence C.},
TITLE = {Classical solutions of fully nonlinear, convex, second-order elliptic equations},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {35},
YEAR = {1982},
NUMBER = {3},
PAGES = {333--363},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {35J60 (93E20)},
MRNUMBER = {0649348},
MRREVIEWER = {Pierre-Louis Lions},
DOI = {10.1002/cpa.3160350303},
ZBLNUMBER = {0469.35022},
}
• [7] L. C. Evans and J. Spruck, "Motion of level sets by mean curvature. I," J. Differential Geom., vol. 33, iss. 3, pp. 635-681, 1991.
@article {7, MRKEY = {1100206},
AUTHOR = {Evans, Lawrence C. and Spruck, J.},
TITLE = {Motion of level sets by mean curvature. {I}},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {33},
YEAR = {1991},
NUMBER = {3},
PAGES = {635--681},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {35K55 (35B05 35D05 53A10)},
MRNUMBER = {1100206},
MRREVIEWER = {Friedrich Sauvigny},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214446559},
ZBLNUMBER = {0726.53029},
}
• [8] M. Gage and R. S. Hamilton, "The heat equation shrinking convex plane curves," J. Differential Geom., vol. 23, iss. 1, pp. 69-96, 1986.
@article {8, MRKEY = {0840401},
AUTHOR = {Gage, M. and Hamilton, R. S.},
TITLE = {The heat equation shrinking convex plane curves},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {23},
YEAR = {1986},
NUMBER = {1},
PAGES = {69--96},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53A04 (35K05 52A40 58E99 58G11)},
MRNUMBER = {0840401},
MRREVIEWER = {R. Osserman},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214439902},
ZBLNUMBER = {0621.53001},
}
• [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second ed., New York: Springer-Verlag, 1983, vol. 224.
@book {9, MRKEY = {0737190},
AUTHOR = {Gilbarg, David and Trudinger, Neil S.},
TITLE = {Elliptic Partial Differential Equations of Second Order},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {224},
EDITION = {Second},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1983},
PAGES = {xiii+513},
ISBN = {3-540-13025-X},
MRCLASS = {35Jxx (35-01)},
MRNUMBER = {0737190},
MRREVIEWER = {O. John},
ZBLNUMBER = {0562.35001},
}
• [10] R. S. Hamilton, "The formation of singularities in the Ricci flow," in Surveys in Differential Geometry, Vol. II, Cambridge, MA, 1995, pp. 7-136.
@inproceedings{10, MRKEY = {1375255},
AUTHOR = {Hamilton, Richard S.},
TITLE = {The formation of singularities in the {R}icci flow},
BOOKTITLE = {Surveys in Differential Geometry, {V}ol. {{\rm II}}},
VENUE={{C}ambridge, {MA},
1993},
PAGES = {7--136},
PUBLISHER = {Internat. Press},
ADDRESS={ Cambridge, MA},
YEAR = {1995},
MRCLASS = {53C21 (58G30)},
MRNUMBER = {1375255},
MRREVIEWER = {Man Chun Leung},
ZBLNUMBER = {0867.53030},
}
• [11] R. S. Hamilton, "Harnack estimate for the mean curvature flow," J. Differential Geom., vol. 41, iss. 1, pp. 215-226, 1995.
@article {11, MRKEY = {1316556},
AUTHOR = {Hamilton, Richard S.},
TITLE = {Harnack estimate for the mean curvature flow},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {41},
YEAR = {1995},
NUMBER = {1},
PAGES = {215--226},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C21 (53A07 58G30)},
MRNUMBER = {1316556},
MRREVIEWER = {Emmanuel Hebey},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214456010},
ZBLNUMBER = {0827.53006},
}
• [12] G. Huisken, "Flow by mean curvature of convex surfaces into spheres," J. Differential Geom., vol. 20, iss. 1, pp. 237-266, 1984.
@article {12, MRKEY = {0772132},
AUTHOR = {Huisken, Gerhard},
TITLE = {Flow by mean curvature of convex surfaces into spheres},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {20},
YEAR = {1984},
NUMBER = {1},
PAGES = {237--266},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C45 (49F05 58F17)},
MRNUMBER = {0772132},
MRREVIEWER = {R. Schneider},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214438998},
ZBLNUMBER = {0556.53001},
}
• [13] G. Huisken, "Local and global behaviour of hypersurfaces moving by mean curvature," in Differential Geometry: Partial Differential Equations on Manifolds, Providence, RI, 1993, pp. 175-191.
@inproceedings{13, MRKEY = {1216584},
AUTHOR = {Huisken, Gerhard},
TITLE = {Local and global behaviour of hypersurfaces moving by mean curvature},
BOOKTITLE = {Differential Geometry: Partial Differential Equations on Manifolds},
VENUE={{L}os {A}ngeles, {CA},
1990},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {54},
PAGES = {175--191},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1993},
MRCLASS = {58E15 (53A10 58G11)},
MRNUMBER = {1216584},
MRREVIEWER = {Li Ma},
ZBLNUMBER = {0791.58090},
}
• [14] G. Huisken and C. Sinestrari, "Mean curvature flow singularities for mean convex surfaces," Calc. Var. Partial Differential Equations, vol. 8, iss. 1, pp. 1-14, 1999.
@article {14, MRKEY = {1666878},
AUTHOR = {Huisken, Gerhard and Sinestrari, Carlo},
TITLE = {Mean curvature flow singularities for mean convex surfaces},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {8},
YEAR = {1999},
NUMBER = {1},
PAGES = {1--14},
ISSN = {0944-2669},
MRCLASS = {58E12 (35K55 53A10)},
MRNUMBER = {1666878},
MRREVIEWER = {John Urbas},
DOI = {10.1007/s005260050113},
ZBLNUMBER = {0992.53052},
}
• [15] G. Huisken and C. Sinestrari, "Convexity estimates for mean curvature flow and singularities of mean convex surfaces," Acta Math., vol. 183, iss. 1, pp. 45-70, 1999.
@article {15, MRKEY = {1719551},
AUTHOR = {Huisken, Gerhard and Sinestrari, Carlo},
TITLE = {Convexity estimates for mean curvature flow and singularities of mean convex surfaces},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {183},
YEAR = {1999},
NUMBER = {1},
PAGES = {45--70},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {53C44 (35K55)},
MRNUMBER = {1719551},
MRREVIEWER = {Ben Andrews},
DOI = {10.1007/BF02392946},
ZBLNUMBER = {0992.53051},
}
• [16] G. Huisken and C. Sinestrari, "Mean curvature flow with surgeries of two-convex hypersurfaces," Invent. Math., vol. 175, iss. 1, pp. 137-221, 2009.
@article {16, MRKEY = {2461428},
AUTHOR = {Huisken, Gerhard and Sinestrari, Carlo},
TITLE = {Mean curvature flow with surgeries of two-convex hypersurfaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {175},
YEAR = {2009},
NUMBER = {1},
PAGES = {137--221},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53C44 (35K93 53C21)},
MRNUMBER = {2461428},
MRREVIEWER = {John Urbas},
DOI = {10.1007/s00222-008-0148-4},
ZBLNUMBER = {1170.53042},
}
• [17] N. Ivochkina, N. Trudinger, and X. Wang, "The Dirichlet problem for degenerate Hessian equations," Comm. Partial Differential Equations, vol. 29, iss. 1-2, pp. 219-235, 2004.
@article {17, MRKEY = {2038151},
AUTHOR = {Ivochkina, Nina and Trudinger, Neil and Wang, Xu-Jia},
TITLE = {The {D}irichlet problem for degenerate {H}essian equations},
JOURNAL = {Comm. Partial Differential Equations},
FJOURNAL = {Communications in Partial Differential Equations},
VOLUME = {29},
YEAR = {2004},
NUMBER = {1-2},
PAGES = {219--235},
ISSN = {0360-5302},
CODEN = {CPDIDZ},
MRCLASS = {35J60 (35J25 35J70)},
MRNUMBER = {2038151},
MRREVIEWER = {Kai Seng Chou},
DOI = {10.1081/PDE-120028851},
ZBLNUMBER = {1140.35418},
}
• [18] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, New York: Springer-Verlag, 1985, vol. 1150.
@book {18, MRKEY = {0810619},
AUTHOR = {Kawohl, Bernhard},
TITLE = {Rearrangements and Convexity of Level Sets in {{\rm PDE}}},
SERIES = {Lecture Notes in Math.},
VOLUME = {1150},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1985},
PAGES = {iv+136},
ISBN = {3-540-15693-3},
MRCLASS = {35-02 (35B50 35J60 49A50)},
MRNUMBER = {0810619},
MRREVIEWER = {Michael Wiegner},
ZBLNUMBER = {0593.35002},
}
• [19] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Dordrecht: D. Reidel Publishing Co., 1987, vol. 7.
@book {19, MRKEY = {0901759},
AUTHOR = {Krylov, N. V.},
TITLE = {Nonlinear Elliptic and Parabolic Equations of the Second Order},
SERIES = {Math. Appl. (Soviet Series)},
VOLUME = {7},
PUBLISHER = {D. Reidel Publishing Co.},
ADDRESS = {Dordrecht},
YEAR = {1987},
PAGES = {xiv+462},
ISBN = {90-277-2289-7},
MRCLASS = {35-02 (35J60 35K55)},
MRNUMBER = {0901759},
ZBLNUMBER = {0619.35004},
}
• [20] N. V. Krylov, "Smoothness of the payoff function for a controllable diffusion process in a domain," Izv. Akad. Nauk SSSR Ser. Mat., vol. 53, iss. 1, pp. 66-96, 1989.
@article {20, MRKEY = {0992979},
AUTHOR = {Krylov, N. V.},
TITLE = {Smoothness of the payoff function for a controllable diffusion process in a domain},
JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
VOLUME = {53},
YEAR = {1989},
NUMBER = {1},
PAGES = {66--96},
ISSN = {0373-2436},
MRCLASS = {93C20 (35K65)},
MRNUMBER = {0992979},
MRREVIEWER = {B. I. Anan{\cprime}ev},
}
• [21] N. V. Krylov, Lectures on fully nonlinear elliptic equations, 1994.
@misc{21,
author={Krylov, N. V.},
TITLE = {Lectures on fully nonlinear elliptic equations},
NOTE={Lipschitz Lectures, Univ. of Bonn},
YEAR={1994},
}
• [22] D. G. Miguel, Differentiation of Integrals in ${{\bf R}}^n$, New York: Springer-Verlag, 1977, vol. 481.
@book{22,
author={Miguel, D. G.},
TITLE={Differentiation of Integrals in ${{\bf R}}^n$},
SERIES={Lecture Notes in Math.},
VOLUME={481},
PUBLISHER={Springer-Verlag},
ADDRESS={New York},
YEAR={1977},
}
• [23] G. Perelman, The entropy formula for the Ricci flow and its geometric applications.
@misc{23,
author={Perelman, G.},
TITLE={The entropy formula for the Ricci flow and its geometric applications},
ARXIV={math.DG/0211159},
}
• [24] G. Perelman, Ricci flow with surgery on three manifolds.
@misc{24,
author={Perelman, G.},
TITLE={Ricci flow with surgery on three manifolds},
ARXIV={math.DG/0303109},
}
• [25] W. Sheng and X. Wang, "Singularity profile in the mean curvature flow," Methods Appl. Anal., vol. 16, iss. 2, pp. 139-155, 2009.
@article {25, MRKEY = {2563745},
AUTHOR = {Sheng, Weimin and Wang, Xu-Jia},
TITLE = {Singularity profile in the mean curvature flow},
JOURNAL = {Methods Appl. Anal.},
FJOURNAL = {Methods and Applications of Analysis},
VOLUME = {16},
YEAR = {2009},
NUMBER = {2},
PAGES = {139--155},
ISSN = {1073-2772},
MRCLASS = {53C44 (35K93)},
MRNUMBER = {2563745},
MRREVIEWER = {Pierre Cardaliaguet},
ZBLNUMBER = {1184.53071},
}
• [26] L. Simon, "The minimal surface equation," in Geometry, V, New York: Springer-Verlag, 1997, vol. 90, pp. 239-272.
@incollection {26, MRKEY = {1490041},
AUTHOR = {Simon, Leon},
TITLE = {The minimal surface equation},
BOOKTITLE = {Geometry, {{\rm V}}},
SERIES = {Encyclopaedia Math. Sci.},
VOLUME = {90},
PAGES = {239--272},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1997},
MRCLASS = {53A10 (35J60 58E12)},
MRNUMBER = {1490041},
MRREVIEWER = {R. Finn},
ZBLNUMBER = {0905.53003},
}
• [27] N. S. Trudinger, "On the Dirichlet problem for Hessian equations," Acta Math., vol. 175, iss. 2, pp. 151-164, 1995.
@article {27, MRKEY = {1368245},
AUTHOR = {Trudinger, Neil S.},
TITLE = {On the {D}irichlet problem for {H}essian equations},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {175},
YEAR = {1995},
NUMBER = {2},
PAGES = {151--164},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {35J65},
MRNUMBER = {1368245},
MRREVIEWER = {John Urbas},
DOI = {10.1007/BF02393303},
ZBLNUMBER = {0887.35061},
}
• [28] N. S. Trudinger and X. Wang, "The Bernstein problem for affine maximal hypersurfaces," Invent. Math., vol. 140, iss. 2, pp. 399-422, 2000.
@article {28, MRKEY = {1757001},
AUTHOR = {Trudinger, Neil S. and Wang, Xu-Jia},
TITLE = {The {B}ernstein problem for affine maximal hypersurfaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {140},
YEAR = {2000},
NUMBER = {2},
PAGES = {399--422},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53A15 (58E15)},
MRNUMBER = {1757001},
MRREVIEWER = {Giandomenico Orlandi},
DOI = {10.1007/s002220000059},
ZBLNUMBER = {0978.53021},
}
• [29] B. White, "The size of the singular set in mean curvature flow of mean-convex sets," J. Amer. Math. Soc., vol. 13, iss. 3, pp. 665-695, 2000.
@article {29, MRKEY = {1758759},
AUTHOR = {White, Brian},
TITLE = {The size of the singular set in mean curvature flow of mean-convex sets},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {13},
YEAR = {2000},
NUMBER = {3},
PAGES = {665--695},
ISSN = {0894-0347},
MRCLASS = {53C44 (49Q20)},
MRNUMBER = {1758759},
MRREVIEWER = {Harold Parks},
DOI = {10.1090/S0894-0347-00-00338-6},
ZBLNUMBER = {0961.53039},
}
• [30] B. White, "The nature of singularities in mean curvature flow of mean-convex sets," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 123-138, 2003.
@article {30, MRKEY = {1937202},
AUTHOR = {White, Brian},
TITLE = {The nature of singularities in mean curvature flow of mean-convex sets},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {16},
YEAR = {2003},
NUMBER = {1},
PAGES = {123--138},
ISSN = {0894-0347},
MRCLASS = {53C44 (49Q20)},
MRNUMBER = {1937202},
MRREVIEWER = {Shu-Yu Hsu},
DOI = {10.1090/S0894-0347-02-00406-X},
ZBLNUMBER = {1027.53078},
}

## Authors

Xu-Jia Wang

Centre for Mathematics and its Applications
The Australian National University
ACT 0200
Australia